Row Space, Column Space, and Nullspace - Lecture Slides | MATH 322, Assignments of Linear Algebra

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Row Space, Column Space, and Nullspace

MATH 322, Linear Algebra I

J. Robert Buchanan

Department of Mathematics

Spring 2007

Introduction

Every matrix has associated with it three vector spaces: row space column space nullspace

Definitions

Definition If A is an m × n matrix, the subspace of R n^ spanned by the row vectors of A is called the row space of A. The subspace of R m spanned by the column vectors of A is called the column space of A. The vector space of solutions to A x = 0 , which is a subspace of R n^ is called the nullspace of A.

Questions: What relationships exist between the row space, column space, and nullspace? What relationships exist between the solutions of A x = b and the row space, column space, and nullspace?

Column Space

If A =

a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n .. .

am 1 am 2 · · · amn

and x =

x 1 x 2 .. . xn

then

A x = x 1

a 11 a 21 .. . am 1

• x 2

a 12 a 22 .. . am 2

• · · · + xn

a 1 n a 2 n .. . amn

which is a linear combination of the columns of A.

Results

Theorem A system of linear equations A x = b is consistent if and only if b lies in the column space of A.

Example Determine if the following system of equations is consistent.  

x 1 x 2 x 3

Results (continued)

Theorem If x 0 is any solution to a consistent nonhomogeneous linear system A x = b and if { v 1 , v 2 ,... , v k } form a basis for the nullspace of A, then every solution of A x = b can be expressed as x = x 0 + c 1 v 1 + c 2 v 2 + · · · + ck v k and conversely for all scalars c 1 , c 2 ,... , ck , the vector x above is a solution to A x = b.

Proof.

General and Particular Solutions

Definition The vector x 0 described previously is called a particular solution to A x = b. The vector

c 1 v 1 + c 2 v 2 + · · · + ck v k

is called a general solution to A x = 0. The general solution to A x = b is is the sum of any particular solution to A x = b and the general solution to A x = 0.

Basis for the Nullspace

Recall: elementary row operations do not change the solution space of a homogeneous linear system A x = 0.

Theorem Elementary row operations do not change the nullspace of a matrix.

Example Find a basis for the nullspace of A where

A =

Basis for the Nullspace

Recall: elementary row operations do not change the solution space of a homogeneous linear system A x = 0.

Theorem Elementary row operations do not change the nullspace of a matrix.

Example Find a basis for the nullspace of A where

A =

Basis for the Row Space

Theorem Elementary row operations do not change the row space of a matrix.

Proof.

Remark: elementary row operations do change the column space of a matrix.

Basis for the Row Space

Theorem Elementary row operations do not change the row space of a matrix.

Proof.

Remark: elementary row operations do change the column space of a matrix.

Results (continued)

Theorem If A and B are row equivalent matrices then: (^1) A given set of column vectors of A is linearly independent if and only if the corresponding column vectors from B are linearly independent. (^2) A given set of column vectors of A forms a basis for the column space of A if and only if the corresponding vectors of B form a basis for the column space of B.

Proof.

Results (continued)

Theorem If a matrix R is in row-echelon form, then the vectors containing the leading 1’s form a basis for the row space and the columns containing the leading 1’s of the row vectors form a basis for the column space.

Example Find bases for the row space and column space of A where

A =

Results (continued)

Theorem If a matrix R is in row-echelon form, then the vectors containing the leading 1’s form a basis for the row space and the columns containing the leading 1’s of the row vectors form a basis for the column space.

Example Find bases for the row space and column space of A where

A =